1. How can I make an automated email send attach an email record to the case that I originated it from?
2. How can I use case variables or Box links to personalize the email sent as part of this automation?
In regards to point one emails sent should show up under the activity history, if you are having an issue with this I would reach out to support. You can use tags in email templates to make them dynamic, the following link will help(https://help.nintex.com/en-US/docgen/docservices/#docgen-sfdc/Services/manageDDP/EmailTemplates.htm).